Abstract Algebra - Size of the generating set

Question

Given a group $G$, say of some size $k$. Can it be shown that it actually has a generating set H whose size can be expressed in terms of $k$? Intuitively, one can say it should be say around $O(\log k)$, as we indeed multiply an element so many times and it begins to repeat after sometime, similar to a Divide and Conquer strategy, where we throw away half the problem at every instance and then move on.

Answer 1

$O(\log k)$ may be an upper bound on the number of generators. But $Z_n$ has only a single generator for any value of $n$, and $D_n$ is generated by just two generators.